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Position, Displacement, and Distance:
Position (x): The location of an object on a number line or coordinate plane.
Displacement (Δx): The change in position from start to end point.
Displacement is a vector (it has direction and magnitude), while distance is a scalar (it only has magnitude, total path length traveled).
Average Velocity vs Instantaneous Velocity
Average Velocity (v avg): 𝑣 𝑎𝑣𝑔=Δ𝑥/Δ𝑡
Instantaneous Velocity (v):
Calculated over an infinitesimally small time interval: 𝑣=𝑑𝑥/𝑑𝑡
(First derivative of position with respect to time)
On a position vs. time graph:
Average velocity = slope of the secant line
Instantaneous velocity = slope of the tangent line at a point
Average Speed vs. Average Velocity
Average speed: Total distance / Total time (ALWAYS positive)
Average velocity: Displacement / Time interval (can be negative if direction is leftward or downward)
Example scenario: A car goes forward 10 m, then backward 10 m.
Average speed ≠ 0, but average velocity = 0.
Uniform Acceleration and Shortcuts
If acceleration is constant:
The instantaneous velocity at the midpoint in time equals the average velocity over the whole time interval
Shortcut Formula:
Derivatives in Kinematics
Instantaneous Velocity: First derivative of position: 𝑣(𝑡)=𝑑𝑥/𝑑𝑡
Instantaneous Acceleration: First derivative of velocity, or second derivative of position: 𝑎(𝑡)=𝑑𝑣/𝑑𝑡=𝑑^2𝑥/𝑑𝑡^2
Example Polynomial Rule:
If:
𝑥(𝑡)=2𝑡^3+3𝑡^2
Then:
𝑣(𝑡)=6𝑡^2+6𝑡
𝑎(𝑡)=12𝑡+6
*KINEMATIC EQUATIONS (For Constant Acceleration Only)
Two-Dimensional Motion: Vectors and Components
Express vectors using unit vector notation:
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